226 Mr. J. Cockle on Differential Covariants. 



Next, consider the differential cubic 



and, to that end, let A denote the operation 



db dc df 



(d \ m d m y 

 -r-j or j-^ and which gi 



to such results as 

 and 



lves rise 



db' _ d d , _ d db _~ 



db ~ db dx' ~ dx db 



>iw= b M)= w - 



Moreover let K 2 and K 3 represent the differential critical func- 

 tions of the second and third order, so that 



K 2 =6 2 -c + Z/, 



K B =2bZ-Zbc+f-V'. 

 Then, first, we see that the operation A reduces each of these cri- 

 tical functions to zero, in other words, that 



AK 2 =0, AK 3 =0: 

 and, in the second place we see that if 



P =35 S 6-(K 8 +K' S ), 



and 



9 =K 2C -(K 3 +K' 2 )&, 

 then also 



A (^£ + ^£ + ^= 0; 



or, in other language, the operator A reduces the expression 



to zero, and that expression is a differential covariant of the given 

 differential cubic. 



Now, introduce the factorial substitution of uv for y, and 

 divide the given cubic and the covariant, just obtained, by u, and 

 denote the respective results by 



(1, B, 0, FX-jL, Ifv 



and 



(k4p,qx|,DV 



